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This might be a rather simple question, but this is troubling me a lot? Please help.

Suppose in a factor analysis model, there are three variables $x_1$, $x_2$ and $x_3$, and two latent factors $w_1$, and $w_2$. Now,

$x_1$ = $w_1$$z_{11}$ + $w_2$$z_{21}$ + $\epsilon_1$

$x_2$ = $w_1$$z_{21}$ + $w_2$$z_{22}$ + $\epsilon_2$

In the above equation, $\epsilon_1$ and $\epsilon_2$ are called unique variances, which are independent of each other. I don't understand how these two are independent and unique, since both $x_1$ and $x_2$ are coming from same latent factors?

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  • $\begingroup$ These are not unique variances in the formulas, but the (unknown) unique variables themselves. See explanation of the factor model: stats.stackexchange.com/a/94104/3277 $\endgroup$
    – ttnphns
    Commented Apr 27, 2016 at 16:18

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The error represents the variance in the measured variables (x1 and x2) which is not explained by the latent factors. The latent factors don't explain all of the variance in the measured variables, only the variance that they share.

If the errors were not unique, that would mean that the variables share some variance that is not explained by the factors. If that were the case, it means that the loadings are not high enough - because there is some additional shared variance that hasn't been explained. The loadings would increase until they soaked up all of the shared variance, and all that was left was unique to each variable.

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